Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#107 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(2,5)(3,7)(9,10,14,20,16,21,15,36)(11,17,25,40,23,38,26,54)(12,27,30,56,32,57,31,60)(13,33,34,44,35,61,28,19)(18,41,42,59,43,58,55,29)(22,45,49,68,47,66,50,71)(24,52,53,65,51,72,46,39)(37,62,64,70,63,69,67,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,22)(13,24)(14,25)(15,26)(16,23)(18,37)(19,39)(20,40)(21,38)(27,45)(28,46)(29,48)(30,49)(31,50)(32,47)(33,52)(34,53)(35,51)(36,54)(41,62)(42,64)(43,63)(44,65)(55,67)(56,68)(57,66)(58,69)(59,70)(60,71)(61,72);(2,5)(6,8)(9,12,28,58,15,31,35,59,16,32,34,41,14,30,13,29)(10,18,33,56,20,42,44,57,21,43,61,60,36,55,19,27)(11,22,46,69,26,50,51,70,23,47,53,62,25,49,24,48)(17,37,52,68,40,64,65,66,38,63,72,71,54,67,39,45);(1,3,4,7)(2,6,5,8)(11,23)(17,38)(22,47)(24,51)(25,26)(37,63)(39,65)(40,54)(45,66)(46,53)(48,70)(49,50)(52,72)(62,69)(64,67)(68,71)Polycyclic presentation:
2^9|a^2=g,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403943);